A number a is a root of P if and only if the polynomial x − a ( of degree one in x ) divides P. It may happen that x − a divides P more than once: if ( x − a )< sup > 2 </ sup > divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that ( x − a )< sup > m </ sup > divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined ( or infinite ) multiplicity.

When n is an integer, the solution P < sub > n </ sub >( x ) that is regular at x = 1 is also regular at x = − 1, and the series for this solution terminates ( i. e. is a polynomial ).

When the same roots occur multiple times, the terms in this formula corresponding to the second and later occurrences of the same root are multiplied by increasing powers of n. For instance, if the characteristic polynomial can be factored as ( x − r )< sup > 3 </ sup >, with the same root r occurring three times, then the solution would take the form

# is a polynomial ( of degree ≤ n − 1 ) in .< ref > Proof: When A is normal, use Lagrange's interpolation formula to construct a polynomial P such that, where are the eigenvalues of A .</ ref >

When the ring is a field, the Cayley – Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial.

When polynomial sequences replace sequences of numbers as images of y < sup > n </ sup > under the linear mapping L, then the umbral method is seen to be an essential component of Rota's general theory of special polynomials, and that theory is the umbral calculus by some more modern definitions of the term.

When the entries of A are nonnegative, however, the permanent can be computed approximately in probabilistic polynomial time, up to an error of εM, where M is the value of the permanent and ε > 0 is arbitrary.

When, with Stirling's or Bessel's, the last term used includes the average of two differences, then one more point is being used than Newton's or other polynomial interpolations would use for the same polynomial degree.

When K is the real numbers, some of the p < sub > i </ sub > might be quadratic, so in the partial fraction decomposition a quotient of a linear polynomial by a power of a quadratic will occur.

When a is a non-positive integer, − n, is a polynomial.

When a is a non-positive integer, − n, is a polynomial.

For univariate algebraic equations these are also called roots, even if, properly speaking, one should say the solutions of the algebraic equation P = 0 are the roots of the polynomial P. When solving an equation, it is important to specify in which set the solutions are allowed.

When at least t out of the n players reveal their points, there is sufficient information to fit a ( t-1 ) th degree polynomial to them, the first coefficient being the secret.

When using the term " quadratic polynomial ", authors sometimes mean " having degree exactly 2 ", and sometimes " having degree at most 2 ".

When the prime is 2, it is conventional to express elements of GF ( p < sup > n </ sup >) as binary numbers, with each term in a polynomial represented by one bit in the corresponding element's binary expression.

When an integer can be represented as a polynomial with small coefficients, the norms that arise are much smaller than those that arise when an integer is represented by a general polynomial.

When the condition number is exactly one, then the algorithm may find an approximation of the solution with an arbitrary precision.

When used in science, for example geology, eras denote clearly defined periods of time of arbitrary but well defined length, such as for example the Mesozoic Era from 252 Ma – 66 Ma, delimited by a start event and an end event.

When the voice of the individual can be heard, the danger of arbitrary interference by the State is greatly reduced.

When the attempt of papal forces to take over the ancient independent Republic of San Marino failed, Clement XII disavowed the arbitrary action of his legate, Cardinal Alberoni, in seizing San Marino, and restored its independence.

When triggered, the note would begin sounding as a noise or an arbitrary low note until the controller logic had tracked the actual pitch, which would be sent by a follow-up message without the need to retrigger the note.

To see this, the three corresponding normal vectors of the corner's perpendicular sides can be considered to form a basis ( a rectangular coordinate system ) ( x, y, z ) in which to represent the direction of an arbitrary incoming ray, b, c. When the ray reflects from the first side, say x, the ray's x component, a, is reversed to-a while the y and z components are unchanged, resulting in a direction of b, c. Similarly, when reflected from side y and finally from side z, the b and c components are reversed.

where C is an arbitrary path connecting the point with zero potential to r. When the curl is zero, the line integral above does not depend on the specific path C chosen but only on its endpoints.

When it is not possible to proceed in an ideal manner, the excavation must be continued in a more arbitrary way, with temporary sections, until discernible stratigraphy is again encountered.

When the Colonies were first settled, " the lawyer was synonymous with the cringing Attorneys-General and Solicitors-General of the Crown and the arbitrary Justices of the King's Court, all bent on the conviction of those who opposed the King's prerogatives, and twisting the law to secure convictions.

When the network topology is a mesh, where nodes are interconnected by fibres to form an arbitrary graph, an additional fibre interconnection device is needed to route the signals from an input port to the desired output port.

When raising an arbitrary matrix ( not necessarily a diagonal matrix ) to a power, it is often helpful to exploit this property by diagonalizing the matrix first.

When Pitt strayed from the Liberal principles of his early days, his brother-in-law severed their political connection and opposed the arbitrary measures which the ministry favoured.

When exactly a canoe can be called a kayak is difficult to determine though, and often arbitrary.

When sample-based synthesis was first developed, most affordable consumer synthesizers could not record arbitrary samples, but instead formed timbres by combining pre-recorded samples from ROM before routing the result through analog or digital filters.

In his major work, Economy and Society he used this term in arbitrary chronological and geographical frames, e. g., he wrote that " this Christian preoccupation with the formulation of dogmas was in Antiquity particularly influenced by the distinctive character of ' intelligentsia ', which was the product of Greek education " When formulating major social classes of his time, Weber combines intelligentsia with other social categories, e. g., he defines a major class consisting of " the propertyless intelligentsia and specialists ( technicians, various kinds of white-collar employees, civil servants – possible with considerable social differences depending on the cost of their training )" and yet other " classes privileged through property and education ".

" When the king acted in an arbitrary and illegal manner he needed the reminder that though he was king over men he was only " God's silly vassal.

When an arbitrary question is asked the " Guardian of Forever ," explains that he is a doorway to any time and place.

When the map φ between manifolds is a diffeomorphism, that is, it has a smooth inverse, then pullback can be defined for the vector fields as well as for 1-forms, and thus, by extension, for an arbitrary mixed tensor field on the manifold.

When the field is an arbitrary function of time t, the polarization is a convolution of the Fourier transform of χ ( ω ) with the E ( t ).

When the U. S. was unable to secure Alvarez's extradition, it paid Sosa, a Mexican national, to kidnap Alvarez and bring him into the U. S. Alvarez claimed that his " arrest " by Sosa was arbitrary because the warrant for his arrest only authorized his arrest within the U. S. The U. S. Court of Appeals for the Ninth Circuit held that Alvarez's abduction constituted arbitrary arrest in violation of international law.

When attempting an attack, it is possible to align duplicate attacks with these intervals ; a NOP slide may be used with shellcode injection, and the string '/ bin / sh ' can be replaced with '//////// bin / sh ' for an arbitrary number of slashes when attempting to return to system.

When the time variable runs over a set which is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a cantor set then one gets dynamic equations on time scales.

When helped by a high degree of structure in lesson presentation, then, and only then, does such a child attain unusual success.

When two places trade with one another, this doctrine supposes that, if the balance be even, neither of them either loses or gains ; but if it leans in any degree to one side, that one of them loses and the other gains in proportion to its declension from the exact equilibrium.

" Venerable / Heroic in Virtue " When enough information has been gathered, the congregation will recommend to the pope that he make a proclamation of the Servant of God's heroic virtue ( that is, that the servant exhibited the theological virtues of faith, hope and charity, and the cardinal virtues of prudence, justice, fortitude and temperance, to a heroic degree ).

When, in the 1970s and 1980s, Jürgen Habermas redefined critical social theory as a theory of communication, i. e. communicative competence and communicative rationality on the one hand, distorted communication on the other, the two versions of critical theory began to overlap or intertwine to a much greater degree than before.

When university degrees became a prerequisite to become a lawyer in England, the degree awarded was the undergraduate LL. B.

When Ahab hears this he repents to such a degree that God relents in punishing Ahab but will punish Jezebel and their son — Ahaziah.

When he completed his B. S., professors Joseph Budd and Louis Pammel convinced Carver to continue at Iowa State for his master's degree.

* When the Legion of Merit is awarded to members of the Armed Forces of the United States it is awarded without reference to degree.

When switching from 700 stadia per degree to 500, he ( or Marinos ) expanded longitude differences between cities accordingly ( a point first realized by P. Gosselin in 1790 ), resulting in serious over-stretching of the Earth's east-west scale in degrees, though not distance.

When contemplating an award of this decoration, the key issue that commanders must take into consideration is the degree to which the enemy caused the injury.

When the matter of honorary degrees came up at the University of Göttingen six years after Germain's death, Gauss lamented, “ proved to the world that even a woman can accomplish something worthwhile in the most rigorous and abstract of the sciences and for that reason would well have deserved an honorary degree.

When Frederick I of Hohenstaufen was chosen as king in 1152, the royal power had been in effective abeyance for twenty-five years, and to a considerable degree, for more than eighty years.

When a sufficient degree of impedance matching is not practical, echo suppressors or echo cancellers, or both, can sometimes reduce the problems.

When it bakes, it becomes crispy but, since it contains somewhat less water, does not expand to the same degree as puff pastry does.

When Advanced Dungeons & Dragons 2nd edition was published, a decision was made not to include angelic or demonic creatures, and so the cosmology was largely ignored, being replaced ( to a certain degree ) by the Spelljammer setting.

As Claire Raines points out in ‘ Beyond Generation X ’, “ never before in history had youth been so idealized as they were at this moment .” When Generation X came along it had much to live up to and to some degree in the shadow of the Boomers, sometimes compared and / or criticized (‘ spoiled ’, ‘ whiners ’ and ‘ the doom generation ’) than not.

When a new fragment of type is discovered, Bayes ' theorem is applied to update the degree of belief for each:

When these results were first reported in 2002, the popular media sensationalized the story and exaggerated the risk, while the manufacturer continued to attempt to minimize the degree of risk.

When Taraki realized the degree of popular dissatisfaction with the reform he began to abandon the policy.

When deficits always resolve between attacks, this is sometimes referred to as benign MS, although people will still accrue some degree of disability in the long term.

When using degree as the units, the arc's length is compared to 1 / 360 of the circumference of the circle.

When he returned home to Bloomington, he decided to finish his law degree at Northwestern University School of Law, attending classes during the week and returning to Bloomington on the weekends to write for the Pantagraph.

When the business cycle is fairly stable, consumer demand surveys and consumer confidence and sentiment indices will often correlate closely and indicate the same direction of the economy, but in times with a high degree of economic or political uncertainty or during a prolonged crisis, the two types of consumer surveys might differ significantly.

* When we write, the ego is active but it is not always active to the same degree.